After the success of her first album, 2008's Youth Novels, Lykke Li went to Los Angeles to write the follow-up. She rented a small cabin in Echo Park and a big piano to put in it, along with an auto-harp and a guitar. "I didn't have a driver's licence so I was stuck on that hill," she says. "I could have gone on for ever, just hanging out, not going on tours."

Sitting in a quiet hotel bar in damp, grey, chilly east London, the 24-year-old Swede looks momentarily wistful. She's just flown in from Stockholm, where she says the weather is even more dreadful, and she seems tired. Her face is pale and her clothes – boots, tights, short skirt, jacket and woollen cape – are uniformly black. She mutters darkly about having to do all these interviews and despising everything she says in them. Another dose of California sun, I feel, would do her good.

"It's like the most exotic place on Earth," she brightens up. "I read this book by David Lynch where he talks about coming to LA from Chicago and he's like, 'There's always this beautiful light – it fills your soul – and the smell of jasmine.' My plan was to take a really long break. But then I wrote all these songs and it felt like if I don't record them they're going to be lost. I'm just going to get too happy [in LA], so I'll just do it, OK?"

She flew back to Stockholm, hooked up with her regular producer Bjorn Yttling, of Peter Björn and John, and took to the studio. Listening to Wounded Rhymes, the album she released last week, you can feel a cold draught of Scandinavian air undercutting the hazy LA heat. It is loose and dreamy with a 60s West Coast vibe – Hammond organs, shoo-wop harmonies – and the sound is more full-bodied than on her first album, but a melancholy chill permeates it. The lyrics throb with loneliness and pain, and – it's one of the song titles – unrequited love.

Was it difficult to make?

"No, it's easy to make a dark album," she laughs. "I love pouring my heart out. People don't want to hear you whine when you're with friends, so you can sing about it instead – it's the best outlet."

The first album was also preoccupied by painful love affairs, though sweetened by cute-sounding hits such as "Little Bit". At the time, she talked about a "really weird relationship" that had been torturing her for years. Is it the same heartbreak haunting her on this album?

She furrows her brow. "The problem is, when I talk about heartbreak or whatever, people want to melt it down to some break-up of a relationship, but it's not about that. If you're a sensitive person, just stepping outside can be heartbreaking."

All this makes Wounded Rhymes sound like a decidedly gloomy prospect, but actually it's a captivating album full of beautiful moments despite – or perhaps because of – Li's unconventional voice, which was childlike on Youth Novels but has become harder and huskier. ("I think I sing like shit," she tells me at one point, though her numerous fans might disagree.) And it's not all melancholy. "Youth Knows No Pain" and "Get Some" are rousing songs with big crashing drums and punchy choruses.

"Like a shotgun/ Needs an outcome/ I'm your prostitute/ You gon' get some," she drawls on the latter track. It sounds very much like a threat, but some have interpreted the line in a less complicated way. What did she mean by it?

"It's my comment on how men, and especially journalists, look on women and write articles about female artists," she says, fixing me with a flinty stare.

I tug at my collar uneasily. Could she explain further?

"How much can you explain? If that's what you think, if that's your opinion, then I'm your prostitute, you're gonna get some. It's not about sex, or being a victim, it's actually really powerful, you know. It's kind of like Scarface or something: 'You want some of this, I'm going to get you some of this.'

"I just want to be free," she goes on. "Men can be whatever they want to be. Like David Bowie. He can have no shirt or go dressed as a woman. Why can't I do that?"

Li doesn't conceal her distaste for the music industry and she has worked hard to insulate herself from it. She releases through her own label, LL Recordings, under licence to Warner Music. Does anyone try to steer her creatively? "Oh God, I could never let that happen. They want me to do interviews and they expect me to show up for concerts." She shakes her head. "God, I actually like it when people try to make me do things. I can fight you know, like waaaah, I can get crazy at one slightest comment."

She says that at the end of her last tour she felt like Muhammad Ali on the stage. I can well believe it. But now, on the brink of another year of touring with the new album, she feels "very shy and vulnerable" at the prospect of performing again. "I'm actually a really shy person," she says. "I just really enjoy being in studio, I don't go out much."

She was born in the south of Sweden but her parents, a musician father and a photographer mother, moved Li and her two siblings around a lot. They lived in Morocco and on a Portuguese mountaintop and spent a lot of time in India – a proper hippie childhood. "It was very free," she says, "but I've seen so much at such a young age and had so much freedom, it can be quite damaging."

In what way?

"You can become quite blasé, and also I have no sense of home, I don't have roots, I've never had that feeling that someone else is going to take care of me, ever. I don't trust people." She laughs, as if trying to soften that last remark. "But I'm a survivor," she says, "so I can take anything."

A few hours later, I go to see her doing an in-store gig at a record shop in east London. She comes out with a male backing band, all dressed in black, and sings a handful of songs from the new album. The audience are appreciative, but quiet. "You're making me nervous," she says, though she appears perfectly composed. Her composure seems, to me, a hard shell, and within it I can imagine Muhammad Ali lurking, ready to punch.

Wounded Rhymes is out now on LL Recordings. Lykke Li tours the UK from 11-19 April